K-uniform Hypergraph Research Articles

The growth rate of multicolor Ramsey numbers of 3-graphs

The q-color Ramsey number of a k-uniform hypergraph G, denoted r(G; q), is the minimum integer N such that any coloring of the edges of the complete k-uniform hypergraph on N vertices contains a monochromatic copy of G. The study of these numbers is one of the most central topics in combinatorics. One natural question, which for triangles goes back to the work of Schur in 1916, is to determine the behavior of r(G; q) for fixed G and q tending to infinity. In this paper, we study this problem for 3-uniform hypergraphs and determine the tower height of r(G; q) as a function of q. More precisely, given a hypergraph G, we determine when r(G; q) behaves polynomially, exponentially or double exponentially in q. This answers a question of Axenovich, Gyárfás, Liu and Mubayi.

Catching a robber on a random k-uniform hypergraph

Abstract The game of Cops and Robber is usually played on a graph, where a group of cops attempt to catch a robber moving along the edges of the graph. The cop number of a graph is the minimum number of cops required to win the game. An important conjecture in this area, due to Meyniel, states that the cop number of an n-vertex connected graph is $O(\sqrt {n})$ . In 2016, Prałat and Wormald showed that this conjecture holds with high probability for random graphs above the connectedness threshold. Moreover, Łuczak and Prałat showed that on a $\log $ -scale the cop number demonstrates a surprising zigzag behavior in dense regimes of the binomial random graph $G(n,p)$ . In this paper, we consider the game of Cops and Robber on a hypergraph, where the players move along hyperedges instead of edges. We show that with high probability the cop number of the k-uniform binomial random hypergraph $G^k(n,p)$ is $O\left (\sqrt {\frac {n}{k}}\, \log n \right )$ for a broad range of parameters p and k and that on a $\log $ -scale our upper bound on the cop number arises as the minimum of two complementary zigzag curves, as opposed to the case of $G(n,p)$ . Furthermore, we conjecture that the cop number of a connected k-uniform hypergraph on n vertices is $O\left (\sqrt {\frac {n}{k}}\,\right )$ .

Linear saturation numbers of Berge-C3 and Berge-C4

The linear saturation number satklin(n,F) (linear extremal number exklin(n,F)) of F is the minimum (maximum) number of hyperedges of an n-vertex linear k-uniform hypergraph containing no member of F as a subgraph, but the addition of any new hyperedge such that the result hypergraph is still a linear k-uniform hypergraph creates a copy of some hypergraph in F. Determining ex3lin(n, Berge-C3) is equivalent to the famous (6,3)-problem, which has been settled in 1976. Since then, determining the linear extremal numbers of Berge cycles was extensively studied. As the counterpart of this problem in saturation problems, the problem of determining the linear saturation numbers of Berge cycles is considered. In this paper, we prove that satklin(n, Berge-Ct)≥⌊n−1k−1⌋ for any integers k≥3, t≥3, and the equality holds if t=3. In addition, we provide an upper bound for sat3lin(n, Berge-C4) and for any disconnected Berge-C4-saturated linear 3-uniform hypergraph, we give a lower bound for the number of hyperedges of it.

A Measure for the Vulnerability of Uniform Hypergraph Networks: Scattering Number

The scattering number of a graph G is defined as s(G)=max{ω(G−X)−|X|:X⊂V(G),ω(G−X)>1}, where X is a cut set of G, and ω(G−X) denotes the number of components in G−X, which can be used to measure the vulnerability of network G. In this paper, we generalize this parameter to a hypergraph to measure the vulnerability of uniform hypergraph networks. Firstly, some bounds on the scattering number are given. Secondly, the relations of scattering number between a complete k-uniform hypergraph and complete bipartite k-uniform hypergraph are discussed.

Incidence Energy of k-Uniform Hypertrees

For a square matrix M, its energy E(M) is the sum of its singular values. Let H be a k-uniform hypergraph, and let B(H) be the incidence matrix of H. The incidence energy BE(H) of H is the energy of B(H). Let T n,d be the set of k-uniform hypertrees of order n and size r with diameter 3 ≤ d ≤ r − 1. In this article, the k-uniform hypertrees with minimum incidence energy over T n,d are characterized. In addition, we have obtained the incidence energy of a hyperstar, and determined which hyperstar has the maximum and minimum incidence energy among all hyperstars with n vertices.

Saturation numbers for Berge cliques

Let F be a graph and H be a hypergraph, both embedded on the same vertex set. We say H is a Berge-F if there exists a bijection ϕ:E(F)→E(H) such that e⊆ϕ(e) for all e∈E(F). We say H is Berge-F-saturated if H does not contain any Berge-F, but adding any missing edge to H creates a copy of a Berge-F. The saturation number satk(n,Berge-F) is the least number of edges in a Berge-F-saturated k-uniform hypergraph on n vertices. We show satk(n,Berge-Kℓ)∼ℓ−2k−1n,for all k,ℓ≥3. Furthermore, we provide some sufficient conditions to imply that satk(n,Berge-F)=O(n) for general graphs F.

A note on rainbow-free colorings of uniform hypergraphs

A rainbow-free coloring of a k-uniform hypergraph H is a vertex-coloring which uses k colors but with the property that no edge of H attains all colors. Koerkamp and Z̆ivný showed that p=(k−1)(logn)/n is the threshold function for the existence of a rainbow-free coloring of the random k-uniform hypergraph Gk(n,p), and presented that the case when p is close to the threshold is open. In this paper, we give an answer to the question.

Edge-colouring graphs with local list sizes

The famous List Colouring Conjecture from the 1970s states that for every graph G the chromatic index of G is equal to its list chromatic index. In 1996 in a seminal paper, Kahn proved that the List Colouring Conjecture holds asymptotically. Our main result is a local generalization of Kahn's theorem. More precisely, we show that, for a graph G with sufficiently large maximum degree Δ and minimum degree δ≥ln25⁡Δ, the following holds: for every assignment L of lists of colours to the edges of G, such that |L(e)|≥(1+o(1))⋅max⁡{deg⁡(u),deg⁡(v)} for each edge e=uv, there is an L-edge-colouring of G. Furthermore, Kahn showed that the List Colouring Conjecture holds asymptotically for linear, k-uniform hypergraphs, and recently Molloy generalized Kahn's original result to correspondence colouring as well as its hypergraph generalization. We prove local versions of all of these generalizations by showing a weighted version that simultaneously implies all of our results.

Dirac-type conditions for spanning bounded-degree hypertrees

We prove that for fixed k, every k-uniform hypergraph on n vertices and of minimum codegree at least n/2+o(n) contains every spanning tight k-tree of bounded vertex degree as a subgraph. This generalises a well-known result of Komlós, Sárközy and Szemerédi for graphs. Our result is asymptotically sharp. We also prove an extension of our result to hypergraphs that satisfy some weak quasirandomness conditions.

A note on the Tuza constant ck for small k

For a hypergraph H, the transversal is a subset of vertices whose intersection with every edge is nonempty. The cardinality of a minimum transversal is the transversal number of H, denoted by τ(H). The Tuza constant ck is defined as sup⁡τ(H)/(m+n), where H ranges over all k-uniform hypergraphs, with m and n being the number of edges and vertices, respectively. We give an upper bound and a lower bound on ck. The upper bound improves the known ones for k≥7, and the lower bound improves the known ones for k∈{7,8,10,11,13,14,17}.

Minimal asymmetric hypergraphs

In this paper, we prove that for any k≥3, there exist infinitely many minimal asymmetric k-uniform hypergraphs. This is in a striking contrast to k=2, where it has been proved recently that there are exactly 18 minimal asymmetric graphs.We also determine, for every k≥1, the minimum size of an asymmetric k-uniform hypergraph.

On the irregularity of uniform hypergraphs

For a k-uniform hypergraph G, let β(G)=Δ(G)−ρ(G), where Δ(G) is the maximum degree of G and ρ(G) is the spectral radius of G via its adjacency tensor. This parameter provides a spectral measure for the irregularity of hypergraphs. We study the β-values of k-uniform hypergraphs with given number of edges. For each of the families of hypertrees, non-power hypertrees, and unicyclic hypergraphs, we determine the first few largest β-values and hypergraphs achieving the values.

Combination of Bases and an Evaluation of the Set of Extremal 3-Uniform Hypergraphs

The main problem with hypergraphs is their storage. If the hypergraph does not contain singularities, then sparse matrices are often used to describe it. Adjacency matrices are often used to work with k-uniform hypergraphs, but they take up a large amount of space in computer memory and, in general, storage of k-uniform arrays is not very convenient. Here we propose one solution for describing and storing extremal k-uniform hypergraphs. This base is a unique characteristic of an extremal hypergraph that uniquely describes it. In addition, bases can be used to search for the power of extremal k-uniform hypergraphs. We present base enumeration algorithms and present a hypothesis about the analytical form of formulas describing the cardinality of the set of extremal k-uniform hypergraphs. For this task, the base combination operation, also introduced here, is used.

Co-degree threshold for rainbow perfect matchings in uniform hypergraphs

Let k and n be two integers, with k≥3, n≡0(modk), and n sufficiently large. We determine the (k−1)-degree threshold for the existence of a rainbow perfect matchings in n-vertex k-uniform hypergraph. This implies the result of Rödl, Ruciński, and Szemerédi on the (k−1)-degree threshold for the existence of perfect matchings in n-vertex k-uniform hypergraphs. In our proof, we identify the extremal configurations of closeness, and consider whether or not the hypergraph is close to the extremal configuration. In addition, we also develop a novel absorbing device and generalize the absorbing lemma of Rödl, Ruciński, and Szemerédi.

The perfect matching and tight Hamilton cycle decomposition of complete n-balanced mk-partite k-uniform hypergraphs

Let r≥k≥2 and Kr×n(k) denote the complete n-balanced r-partite k-uniform hypergraph, whose vertex set consists of r parts, each part has n vertices, and whose edge set contains all the k-element subsets with no two vertices from one part. A Hamilton cycle decomposition of Kr×n(k) is a partition of the edge set E(Kr×n(k)) into Hamilton cycles. In this paper, we consider the perfect matching decomposition and the tight Hamilton cycle decomposition of Kmk×n(k) for m≥2. We obtain the following results. (1) Let k≥3, m≥1 and n≥1. Then Kmk×n(k) has a perfect matching decomposition. (2) If Kmk(k) has a Hamilton cycle decomposition, then Kmk×n(k) has a tight Hamilton cycle decomposition.

Hypergraph partitioning using tensor eigenvalue decomposition.

Hypergraphs have gained increasing attention in the machine learning community lately due to their superiority over graphs in capturing super-dyadic interactions among entities. In this work, we propose a novel approach for the partitioning of k-uniform hypergraphs. Most of the existing methods work by reducing the hypergraph to a graph followed by applying standard graph partitioning algorithms. The reduction step restricts the algorithms to capturing only some weighted pairwise interactions and hence loses essential information about the original hypergraph. We overcome this issue by utilizing tensor-based representation of hypergraphs, which enables us to capture actual super-dyadic interactions. We extend the notion of minimum ratio-cut and normalized-cut from graphs to hypergraphs and show that the relaxed optimization problem can be solved using eigenvalue decomposition of the Laplacian tensor. This novel formulation also enables us to remove a hyperedge completely by using the "hyperedge score" metric proposed by us, unlike the existing reduction approaches. We propose a hypergraph partitioning algorithm inspired from spectral graph theory and also derive a tighter upper bound on the minimum positive eigenvalue of even-order hypergraph Laplacian tensor in terms of its conductance, which is utilized in the partitioning algorithm to approximate the normalized cut. The efficacy of the proposed method is demonstrated numerically on synthetic hypergraphs generated by stochastic block model. We also show improvement for the min-cut solution on 2-uniform hypergraphs (graphs) over the standard spectral partitioning algorithm.

A jump to the Narayana number for hereditary properties of ordered 3-uniform hypergraphs

For k,l≥2 we consider ideals of edge l-colored complete k-uniform hypergraphs (n,χ) with vertex sets [n]={1,2,…n} for n∈N. An ideal is a set of such colored hypergraphs that is closed under the induced ordered sub-hypergraph relation. We obtain analogues of two results of Klazar (2010) who considered graphs, namely we prove two jumps for growth functions of such ideals of colored hypergraphs. The first jump is for any k,l≥2 and says that the growth function is either eventually constant or at least n−k+2. The second jump is only for k=3 and l=2 and says that the growth function of an ideal of edge two-colored complete 3-uniform hypergraphs grows either at most polynomially, or for n≥23 at least as Gn where Gn is the sequence defined by G1=G2=1, G3=2 and Gn=Gn−1+Gn−3 for n≥4. The lower bounds in both jumps are tight.

Counting Hamilton Cycles in Dirac Hypergraphs

For 0le ell <k, a Hamilton ell -cycle in a k-uniform hypergraph H is a cyclic ordering of the vertices of H in which the edges are segments of length k and every two consecutive edges overlap in exactly ell vertices. We show that for all 0le ell <k-1, every k-graph with minimum co-degree delta n with delta >1/2 has (asymptotically and up to a subexponential factor) at least as many Hamilton ell -cycles as a typical random k-graph with edge-probability delta . This significantly improves a recent result of Glock, Gould, Joos, Kühn and Osthus, and verifies a conjecture of Ferber, Krivelevich and Sudakov for all values 0le ell <k-1.

Saturation for the 3-uniform loose 3-cycle

Let F and H be k-uniform hypergraphs. We say H is F-saturated if H does not contain a subgraph isomorphic to F, but H+e does for any hyperedge e∉E(H). The saturation number of F, denoted satk(n,F), is the minimum number of edges in an F-saturated k-uniform hypergraph H on n vertices. Let C3(3) denote the 3-uniform loose cycle on 3 edges. In this work, we prove that(43+o(1))n≤sat3(n,C3(3))≤32n+O(1). This is the first non-trivial result on the saturation number for a fixed short hypergraph cycle.

OPTIMIST: Lightweight and Transparent IDS With Optimum Placement Strategy to Mitigate Mixed-Rate DDoS Attacks in IoT Networks

Distributed denial of service (DDoS) attacks are widespread for Internet of things (IoT) systems that aim to disrupt the availability of a system completely (high-rate DDoS) or partially (low-rate DDoS). Design and placement of Intrusion Detection Systems (IDS) for DDoS attacks on IoT systems are challenging due to the low power and lossy nature of networks. Existing IDSs are designed to handle either high-rate or low-rate DDoS but cannot handle both with good accuracy. Existing IDS placement techniques are mostly non-transparent, making malicious nodes aware of the presence of IDS nodes. Most of the IDS placement strategies are non-optimal, making them energy inefficient. Accordingly, this work proposes a transparent, optimally placed, distributed IDS solution, namely OPTIMIST, which can handle both high-rate and low-rate DDoS attacks with good accuracy. The placement problem is formulated as the weighted minimum vertex cover problem of a K-uniform hypergraph and solved with an approximation algorithm. The IDS module is based on a LSTM model where a novel offline training method for LSTM is proposed using WGAN-generated artificial flows. Extensive experimentation on simulation and testbed shows that the OPTIMIST can best achieve the balance between DDoS detection and energy overhead.